L(s) = 1 | + 2.43·2-s − 0.146·3-s + 3.91·4-s − 5-s − 0.355·6-s + 0.709·7-s + 4.66·8-s − 2.97·9-s − 2.43·10-s + 5.65·11-s − 0.572·12-s − 3.65·13-s + 1.72·14-s + 0.146·15-s + 3.51·16-s + 6.54·17-s − 7.24·18-s + 7.26·19-s − 3.91·20-s − 0.103·21-s + 13.7·22-s − 1.55·23-s − 0.681·24-s + 25-s − 8.89·26-s + 0.874·27-s + 2.77·28-s + ⋯ |
L(s) = 1 | + 1.72·2-s − 0.0844·3-s + 1.95·4-s − 0.447·5-s − 0.145·6-s + 0.267·7-s + 1.64·8-s − 0.992·9-s − 0.769·10-s + 1.70·11-s − 0.165·12-s − 1.01·13-s + 0.460·14-s + 0.0377·15-s + 0.877·16-s + 1.58·17-s − 1.70·18-s + 1.66·19-s − 0.875·20-s − 0.0226·21-s + 2.93·22-s − 0.323·23-s − 0.139·24-s + 0.200·25-s − 1.74·26-s + 0.168·27-s + 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.815682512\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.815682512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.146T + 3T^{2} \) |
| 7 | \( 1 - 0.709T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 - 7.26T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 8.99T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 0.705T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 + 6.84T + 67T^{2} \) |
| 71 | \( 1 + 1.54T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 1.25T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.283102921196717794975302883379, −8.031536536549195444471789083900, −7.42536910944691736193469075932, −6.46123246777971119381274704173, −5.82775460568138534240688446131, −5.03419907605505647758410412837, −4.32189356242777469666841257636, −3.34872557615557408384124473566, −2.83184479180538261256553376375, −1.27656467384826865339052534287,
1.27656467384826865339052534287, 2.83184479180538261256553376375, 3.34872557615557408384124473566, 4.32189356242777469666841257636, 5.03419907605505647758410412837, 5.82775460568138534240688446131, 6.46123246777971119381274704173, 7.42536910944691736193469075932, 8.031536536549195444471789083900, 9.283102921196717794975302883379